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We present observations of SN 2015bn (= PS15ae = CSS141223-113342+004332 = MLS150211-113342+004333), a Type I superluminous supernova (SLSN) at redshift $z=0.1136$. As well as being one of the closest SLSNe I yet discovered, it is intrinsically brighter ($M_U\approx-23.1$) and in a fainter galaxy ($M_B\approx-16.0$) than other SLSNe at $z\sim0.1$. We used this opportunity to collect the most extensive dataset for any SLSN I to date, including densely-sampled spectroscopy and photometry, from the UV to the NIR, spanning $-$50 to +250 days from optical maximum. SN 2015bn fades slowly, but exhibits surprising undulations in the light curve on a timescale of 30-50 days, especially in the UV. The spectrum shows extraordinarily slow evolution except for a rapid transformation between +7 and +20-30 days. No narrow emission lines from slow-moving material are observed at any phase. We derive physical properties including the bolometric luminosity, and find slow velocity evolution and non-monotonic temperature and radial evolution. A deep radio limit rules out a healthy off-axis gamma-ray burst, and places constraints on the pre-explosion mass loss. The data can be consistently explained by a $\gtrsim10\,{\rm M}_\odot$ stripped progenitor exploding with $\sim 10^{51}\,$erg kinetic energy, forming a magnetar with a spin-down timescale of $\sim20$ days (thus avoiding a gamma-ray burst) that reheats the ejecta and drives ionization fronts. The most likely alternative scenario -- interaction with $\sim20\,{\rm M}_\odot$ of dense, inhomogeneous circumstellar material -- can be tested with continuing radio follow-up., Slightly updated to match published version